Question

How do I simplify $\left( 5{{x}^{2}} \right)\left( 2{{x}^{5}} \right)?$

Answer 1

Hint: Here we use polynomial multiplication. We multiply the coefficients to get the new coefficient. Also, we multiply the variables regardless of their exponents.

Complete step by step solution:
Consider the given polynomial expression $\left( 5{{x}^{2}} \right)\left( 2{{x}^{5}} \right)$
Suppose that we have the polynomial expression $\left( ax \right)\left( b \right),$ we simplify it as $abx.$
That is, $\left( ax \right)\left( b \right)=axb.$
We know the identity which says ${{x}^{n}}\cdot {{x}^{m}}={{x}^{n+m}}$
Also, we can do the following multiplication $\left( a{{x}^{n}} \right)\left( b{{x}^{m}} \right)=ab{{x}^{n+m}}$
According to these identities and rules, we can multiply the coefficients and also the variables regardless of their exponents.
Now the coefficients of the variables in the polynomial expression are $5$ and $2.$
Now we multiply these coefficients to get the coefficient of the variable in the product polynomial expression.
So, the coefficient in the product polynomial expression is $5\times 2=10.$
The exponent of the first variable is $2$ and the exponent of the second exponent is $5.$
When we use the identity ${{x}^{n}}\cdot {{x}^{m}}={{x}^{n+m}},$ we get ${{x}^{2}}\cdot {{x}^{5}}={{x}^{2+5}}={{x}^{7}}.$
So, the new product polynomial is $10{{x}^{7}}.$
When we use $\left( a{{x}^{n}} \right)\left( b{{x}^{m}} \right)=ab{{x}^{n+m}},$ the given polynomial expression becomes the following
$\Rightarrow \left( 5{{x}^{2}} \right)\left( 2{{x}^{5}} \right)=\left( 5\times 2 \right){{x}^{2+5}}$
And from this, we will get
$\Rightarrow \left( 5{{x}^{2}} \right)\left( 2{{x}^{5}} \right)=10{{x}^{7}}.$
Hence, the simplified form of the given polynomial expression is given by $\left( 5{{x}^{2}} \right)\left( 2{{x}^{5}} \right)=10{{x}^{7}}.$

Note:
In a simple way, we can solve the given polynomial expression as:
Consider the given polynomial expression $\left( 5{{x}^{2}} \right)\left( 2{{x}^{5}} \right).$
We know that ${{x}^{n}}=x\cdot x\cdot x\cdot ...\cdot {{x}_{n \text{times}}}.$
So, we will get ${{x}^{2}}=x\cdot x.$
Also, we can write ${{x}^{5}}=x\cdot x\cdot x\cdot x\cdot x.$
Now we get the given polynomial expression as
$\Rightarrow \left( 5{{x}^{2}} \right)\left( 2{{x}^{5}} \right)=\left( 5\cdot x\cdot x \right)\left( 2\cdot x\cdot x\cdot x\cdot x\cdot x \right).$
We can approach this equation in two different ways.
The first one is to multiply all the terms – the coefficient-coefficient multiplication and the variable multiplications,
$\Rightarrow \left( 5{{x}^{2}} \right)\left( 2{{x}^{5}} \right)=5\cdot 2\cdot \left( x\cdot x \right)\left( x\cdot x \right)x\cdot x\cdot x=5\cdot 2\cdot {{x}^{2}}\cdot {{x}^{2}}\cdot {{x}^{3}}=10{{x}^{2+2+3}}=10{{x}^{7}}.$
The second method is the easier one. We just multiply the coefficients, then count the number of $x$ int the equation and put that number as the exponent of $x.$
In the equation, the product of coefficients is $10$ and the number of $x=7.$
Therefore, the solution is $\left( 5{{x}^{2}} \right)\left( 2{{x}^{5}} \right)=10{{x}^{7}}.$